Abstract
Symmetry of information establishes a relation between the information that x has about y (denoted I(x : y)) and the information that y has about x (denoted I(y : x)). In classical information theory, the two are exactly equal, but in algorithmical information theory, there is a small excess quantity of information that differentiates the two terms, caused by the necessity of packaging information in a way that makes it accessible to algorithms. It was shown in [Zim11] that in the case of strings with simple complexity (that is the Kolmogorov complexity of their Kolmogorov complexity is small), the relevant information can be packed in a very economical way, which leads to a tighter relation between I(x : y) and I(y : x) than the one provided in the classical symmetry-of-information theorem of Kolmogorov and Levin. We give here a simpler proof of this result.
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References
Chang, C., Lyuu, Y., Ti, Y., Shen, A.: Sets of k-independent sets. International Journal of Foundations of Computer Science 21(3), 321–327 (2010)
Downey, R., Hirschfeldt, D.: Algorithmic randomness and complexity. Springer, Heidelberg (2010)
Fortnow, L., Hitchcock, J.M., Pavan, A., Vinodchandran, N.V., Wang, F.: Extracting Kolmogorov Complexity with Applications to Dimension Zero-One Laws. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 335–345. Springer, Heidelberg (2006)
Hitchcock, J., Pavan, A., Vinodchandran, N.V.: Kolmogorov complexity in randomness extraction. In: Electronic Colloquium on Computational Complexity (ECCC), pp. 09–071 (2009)
Li, M., Vitanyi, P.: An introduction to Kolmogorov complexity and its applications, 3rd edn. Springer, Heidelberg (2008); 1st edn. (1993)
Zimand, M.: Counting Dependent and Independent Strings. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 689–700. Springer, Heidelberg (2010)
Zimand, M.: Possibilities and impossibilities in Kolmogorov complexity extraction. SIGACT News 41(4) (December 2010)
Zimand, M.: Symmetry of information and bounds on nonuniform randomness extraction via Kolmogorov complexity. In: Proceedings 26th IEEE Conference on Computational Complexity, pp. 148–156 (June 2011)
Zvonkin, A., Levin, L.: The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms. Russian Mathematical Surveys 25(6), 83–124 (1970)
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Zimand, M. (2012). Symmetry of Information: A Closer Look. In: Dinneen, M.J., Khoussainov, B., Nies, A. (eds) Computation, Physics and Beyond. WTCS 2012. Lecture Notes in Computer Science, vol 7160. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27654-5_18
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DOI: https://doi.org/10.1007/978-3-642-27654-5_18
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